# MATH 1130: Show Using an Honest Hilbert Proof thatΒ β’π΄β§(π΄β¨π΅)β‘π΄: Discrete Mathematics 1 Assignment, DC, Canada

University | Douglas College (DC) |

Subject | MATH 1130: Discrete Mathematics 1 |

## Question 1:

Show using an honest Hilbert proof thatΒ β’π΄β§(π΄β¨π΅)β‘π΄.

For this question, your answer must be a Hilbert proof according to Definition 1.4.5, no shortcuts (including meta theorems or absolute theorems from class/book) are allowed.

## Question 2:

(a) Prove that for any wffs π΄,π΅,πΆ, D and any propositional variable π,

π΄β(π΅β‘πΆ)β’π΄β(π·[π:=π΅]β‘π·[π:=πΆ])

(Hint: Using the deduction theorem or Post’s theorem makes this easier, although there are other proofs).

(b) Is it true thatΒ π΄β(π΅βπΆ)β’π΄β(π·[π:=π΅]βπ·[π:=πΆ])? If yes, give a proof of this, and if no, find a counterexample.

## Question 3:

Consider the stringΒ ((((πβ¨π)β‘π)β((πβ§(Β¬π))β‘π))βπ)

(a) Write a formula-calculation to show this string is a wff.

(b) Is this formula a tautology? Explain why or why not.

#### Hire a writer to get plagiarism free assignment answers of this question

info@studentsassignmenthelp.com## Question 4:

(a) Give an example of a well-formed formulaΒ π΄Β so that we do NOT haveΒ β’Β¬π΄. Explain why your answer works (using soundness might help).

(b) What is wrong with the following equational proof ofΒ β’Β¬π΄?

Β¬π΄

ββΒ (Β¬-introduction axiom)

π΄β‘β₯

ββΒ (Leibniz from prev. line, “C-part”Β πβ‘β₯)

β₯β‘β₯

ββΒ (β€Β vs.Β β₯)

β€

## Question 5:

Use the method we discussed from the “Weak Post’s Theorem” notes to show:

β’((πβ¨Β¬πβ¨π)β§(πβπ))β(πβπ)

## Question 6:

Write an equational-style proof for the following: πβ(πβ¨π)β’(πβ¨(πβ¨π))β(πβ¨π).

## Question 7:

Give proof to show

β’((πβπ)β§(πβπ))β(πβ(πβ§π))

(Use equational or Hilbert proof styles. Post’s theorem is not allowed).

## Question 8:

What is wrong with the following equational proof ofΒ β’(π΄βπΆ)β‘π΄?

π΄βπΆ

ββΒ (implication axiom)

π΄β¨πΆβ‘πΆ

ββΒ (Leibniz+redundant true; “C-part”Β π΄β¨π)

π΄β¨β€

ββΒ (β¨-identity)

We provide **online math assignment help** to Douglas College students at a very reliable price. Our (MATH 1130) **Discrete Mathematics assignment writing** services are available 24/7 and our writers are ready to assist you any time in Canada. We boast a pool of experienced and highly skilled discrete mathematics **assignment writers**.